@Michael – re your first question, we're assuming here that \$$a, b, c, d\$$ are all mutually distinct (there's no "collapsing" going on).

If you set \$$a = I\$$ then you can prove that \$$c\otimes d \leq c\$$ and \$$c\otimes d \leq d\$$.

But that means \$$c\otimes d\$$ is a common lower bound of \$$c\$$ and \$$d\$$ – and no such lower bound exists.

Intuitively I think the best way of looking at this is that the partial order is too irregular to admit a monoidal structure: \$$a\$$ and \$$b\$$ don't have a join, \$$c\$$ and \$$d\$$ don't have a meet, there is no top element, nor is there a bottom one... basically \$$\leq\$$ is too badly behaved for any \$$\otimes\$$ operation to respect it.